11.4 Three-Dimensional Figures
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1 11. Three-Dimensional Figures Essential Question What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? A polyhedron is a solid that is bounded by polygons, called faces. Each vertex is a point. Each edge is a segment of a line. Each face is a portion of a plane. edge face vertex Analyzing a Property of Polyhedra Work with a partner. The five Platonic solids are shown below. Each of these solids has congruent regular polygons as faces. Complete the table by listing the numbers of vertices, edges, and faces of each Platonic solid. tetrahedron cube octahedron dodecahedron icosahedron Solid Vertices, V Edges, E Faces, F tetrahedron cube octahedron CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to reason inductively about data. dodecahedron icosahedron Communicate Your Answer. What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? (Note: Swiss mathematician Leonhard Euler ( ) discovered a formula that relates these quantities.). Draw three polyhedra that are different from the Platonic solids given in Exploration 1. Count the numbers of vertices, edges, and faces of each polyhedron. Then verify that the relationship you found in Question is valid for each polyhedron. Section 11. Three-Dimensional Figures 617
2 11. Lesson What You Will Learn Core Vocabulary polyhedron, p. 61 face, p. 61 edge, p. 61 vertex, p. 61 cross section, p. 619 solid of revolution, p. 60 axis of revolution, p. 60 Previous solid prism pyramid cylinder cone sphere base Classify solids. Describe cross sections. Sketch and describe solids of revolution. Classifying Solids A three-dimensional figure, or solid, is bounded by flat or curved surfaces that enclose a single region of space. A polyhedron is a solid that is bounded by polygons, called faces. An edge of a polyhedron is a line segment formed by the intersection of two faces. A vertex of a polyhedron is a point where three or more edges meet. The plural of polyhedron is polyhedra or polyhedrons. Core Concept Types of Solids Polyhedra vertex Not Polyhedra edge face prism cylinder cone pyramid sphere Pentagonal prism Bases are pentagons. To name a prism or a pyramid, use the shape of the base. The two bases of a prism are congruent polygons in parallel planes. For example, the bases of a pentagonal prism are pentagons. The base of a pyramid is a polygon. For example, the base of a triangular pyramid is a triangle. Triangular pyramid Base is a triangle. Classifying Solids Tell whether each solid is a polyhedron. If it is, name the polyhedron. a. b. c. SOLUTION a. The solid is formed by polygons, so it is a polyhedron. The two bases are congruent rectangles, so it is a rectangular prism. b. The solid is formed by polygons, so it is a polyhedron. The base is a hexagon, so it is a hexagonal pyramid. c. The cone has a curved surface, so it is not a polyhedron. 61 Chapter 11 Circumference, Area, and Volume
3 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the solid is a polyhedron. If it is, name the polyhedron Describing Cross Sections Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section. For example, three different cross sections of a cube are shown below. square rectangle triangle Describing Cross Sections Describe the shape formed by the intersection of the plane and the solid. a. b. c. d. e. f. SOLUTION a. The cross section is a hexagon. b. The cross section is a triangle. c. The cross section is a rectangle. d. The cross section is a circle. e. The cross section is a circle. f. The cross section is a trapezoid. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Describe the shape formed by the intersection of the plane and the solid Section 11. Three-Dimensional Figures 619
4 Sketching and Describing Solids of Revolution A solid of revolution is a three-dimensional figure that is formed by rotating a two-dimensional shape around an axis. The line around which the shape is rotated is called the axis of revolution. For example, when you rotate a rectangle around a line that contains one of its sides, the solid of revolution that is produced is a cylinder. Sketching and Describing Solids of Revolution Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid. a. 9 b. 9 SOLUTION a. 9 b. The solid is a cylinder with a height of 9 and a base radius of. The solid is a cone with a height of and a base radius of. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid Chapter 11 Circumference, Area, and Volume
5 11. Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. VOCABULARY A(n) is a solid that is bounded by polygons.. WHICH ONE DOESN T BELONG? Which solid does not belong with the other three? Explain your reasoning. Monitoring Progress and Modeling with Mathematics In Exercises 6, match the polyhedron with its name... In Exercises 11 1, describe the cross section formed by the intersection of the plane and the solid. (See Example.) A. triangular prism B. rectangular pyramid C. hexagonal pyramid D. pentagonal prism In Exercises 7 10, tell whether the solid is a polyhedron. If it is, name the polyhedron. (See Example 1.) 7.. In Exercises 1 1, sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid. (See Example.) Section 11. Three-Dimensional Figures 61
6 19. ERROR ANALYSIS Describe and correct the error in identifying the solid. The solid is a rectangular pyramid.. ATTENDING TO PRECISION The figure shows a plane intersecting a cube through four of its vertices. The edge length of the cube is 6 inches. 0. HOW DO YOU SEE IT? Is the swimming pool shown a polyhedron? If it is, name the polyhedron. If not, explain why not. a. Describe the shape of the cross section. b. What is the perimeter of the cross section? c. What is the area of the cross section? REASONING In Exercises 9, tell whether it is possible for a cross section of a cube to have the given shape. If it is, describe or sketch how the plane could intersect the cube. 9. circle 0. pentagon 1. rhombus. isosceles triangle. hexagon. scalene triangle In Exercises 1 6, sketch the polyhedron. 1. triangular prism. rectangular prism. pentagonal prism. hexagonal prism. square pyramid 6. pentagonal pyramid 7. MAKING AN ARGUMENT Your friend says that the polyhedron shown is a triangular prism. Your cousin says that it is a triangular pyramid. Who is correct? Explain your reasoning.. REASONING Sketch the composite solid produced by rotating the figure around the given axis. Then identify and describe the composite solid. a. b THOUGHT PROVOKING Describe how Plato might have argued that there are precisely five Platonic Solids (see page 617). (Hint: Consider the angles that meet at a vertex.) Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Decide whether enough information is given to prove that the triangles are congruent. If so, state the theorem you would use. (Sections.,., and.6) 7. ABD, CDB. JLK, JLM 9. RQP, RTS A B J Q R S D C K L M P T 6 Chapter 11 Circumference, Area, and Volume
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